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34 {\LARGE {\bf Materials Physics II}\\}
39 {\Large\bf Tutorial 1 - proposed solutions}
42 \section{Indirect band gap of silicon}
46 \item Photon wavelength:\\
47 $E_g=\hbar\omega=\hbar\frac{2\pi}{T}=\hbar 2\pi v
48 \stackrel{c=v\lambda}{=}\hbar 2\pi\frac{c}{\lambda}$
49 $\Rightarrow \lambda=\frac{\hbar 2\pi c}{E_g}
50 =\frac{hc}{E_g}=\ldots=1.11 \, \mu m$
51 \item Photon momentum:\\
52 $p=\hbar k=\hbar\frac{2\pi}{\lambda}=\frac{h}{\lambda}
53 =\ldots=5.97 \cdot 10^{-28} \, kg\frac{m}{s}$
55 \item Phonon momentum necessary for transition:\\
56 $\Delta p=\hbar \cdot \Delta k=\hbar \cdot 0.85 \, \frac{2\pi}{a}
57 =\frac{0.85 \, h}{a}=\ldots=1.04 \cdot 10^{-24} \, kg\frac{m}{s}$\\
58 $\rightarrow$ Phonon momentum 3 orders of magnitude below
59 the momentum necessary for transition!
61 \item Photon momentum insufficient.
62 Momentum contribution of phonon (lattice vibration) required.\\
63 $\Rightarrow$ Probability of transition very small.
64 \item Recombination energy of electron-hole pairs most probably
65 released as vibrational energy of the lattice.\\
66 $\Rightarrow$ Only direct band gap semiconductors suitable for
67 effective photon generation.
71 \section{Charge carrier density of semiconductors}
74 \item Calculation of $n$:\\
75 $\forall \epsilon$ of states within conduction band:
76 $\epsilon-\mu_{\text{F}} >> k_{\text{B}}T$
79 \frac{1}{\exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})+1}\approx
80 \exp(-\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})$\\
81 Parabolic approximation:
82 $D_c(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}(\epsilon-E_c)^{1/2}$
84 $n=\int_{E_{\text{c}}}^{\infty}D_c(\epsilon)f(\epsilon,T)d\epsilon\approx
85 \frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}
86 \exp(\frac{\mu_{\text{F}}}{k_{\text{B}}T})
87 \int_{E_{\text{c}}}^{\infty}(\epsilon-E_c)^{1/2}
88 \exp(-\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
89 Use: $x=(\epsilon - E_{\text{c}})/k_{\text{B}}T$
90 $\Rightarrow\epsilon=xk_{\text{B}}T+E_{\text{c}}$ and
91 $d\epsilon=k_{\text{B}}Tdx$\\
93 $n=\frac{1}{2\pi^2}(\frac{2m_nk_{\text{B}}T}{\hbar^2})^{3/2}
94 \exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})
95 \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=\frac{\sqrt{\pi}}{2}}=
96 \underbrace{2(\frac{m_nk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{c}}}
97 \exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})=
98 N_{\text{c}}\exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})$
99 \item In the same way, calculate $p$:\\
100 $\forall \epsilon$ of states within conduction band:
101 $\mu_{\text{F}}-\epsilon >> k_{\text{B}}T$
104 1-\frac{1}{\exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})+1}\approx
105 \exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})$\\
106 Parabolic approximation:
107 $D_v(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}(E_v-\epsilon)^{1/2}$
109 $p=\int_{-\infty}^{E_{\text{v}}}D_v(\epsilon)(1-f(\epsilon,T))d\epsilon\approx
110 \frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}
111 \exp(-\frac{\mu_{\text{F}}}{k_{\text{B}}T})
112 \int_{-\infty}^{E_{\text{v}}}(E_v-\epsilon)^{1/2}
113 \exp(\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
114 Use: $x=(E_{\text{v}}-\epsilon)/k_{\text{B}}T$
115 $\Rightarrow\epsilon=E_{\text{v}}-xk_{\text{B}}T$ and
116 $d\epsilon={\color{red}-}k_{\text{B}}Tdx$\\
118 $p=\frac{1}{2\pi^2}(\frac{2m_pk_{\text{B}}T}{\hbar^2})^{3/2}
119 \exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})
120 \underbrace{\int_{{\color{red}0}}^{{\color{red}\infty}}x^{1/2}e^{-x}dx}_{=\frac{\sqrt{\pi}}{2}}=
121 \underbrace{2(\frac{m_pk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{v}}}
122 \exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})=
123 N_{\text{v}}\exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})$